We consider the semiclassical Schrödinger equation on a compact negatively curved surface. For any sequence of initial data microlocalized on the unit cotangent bundle, we look at the quantum evolution (below the Ehrenfest time) under small perturbations of the Schrödinger equation, and we prove that, in the semiclassical limit, and for typical perturbations, the solutions become equidistributed on the unit cotangent bundle.

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

We study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.

In this note, I wish to describe the first order semiclassical approximation to thespectrum of one frequency quasi-periodic operators. In the case of a sampling functionwith two critical points, the spectrum exhibits two gaps in the leading orderapproximation. Furthermore, I will give an example of a two frequency quasi-periodicoperator, which has no gaps in the leading order of the semiclassical approximation.